For questions about recurrence relations, convergence tests, and identifying sequences. For questions on finite sums use the (summation) instead.

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5
votes
10answers
471 views

Why does $\sum_{n = 0}^\infty \frac{n}{2^n}$ converge to 2? [duplicate]

Apparently, $$ \sum_{n = 0}^\infty \frac{n}{2^n} $$ converges to 2. I'm trying to figure out why. I've tried viewing it as a geometric series, but it's not quite a geometric series since the ...
5
votes
2answers
2k views

Show $\sum\limits_{n=0}^{\infty}{2n \choose n}x^n=(1-4x)^{-1/2}$

How do you prove that $\sum\limits_{n=0}^{\infty}{2n \choose n}x^n=(1-4x)^{-1/2}$? I tried to identify the sum as a binomial series, but the $4$ and the $-1/2$ puzzle me. (This series arises in ...
95
votes
4answers
3k views

Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24}$

I would like to prove that $\displaystyle\sum_{\substack{n=1\\n\text{ odd}}}^{\infty}\frac{n}{e^{n\pi}+1}=\frac1{24}$. I found a solution by myself 10 hours after I posted it, here it is: ...
36
votes
1answer
3k views

Convergence of $\sum \limits_{n=1}^{\infty}\sin(n^k)/n$

Does $S_k= \sum \limits_{n=1}^{\infty}\sin(n^k)/n$ converge for all $k>0$? Motivation: I recently learned that $S_1$ converges. I think $S_2$ converges by the integral test. Was the question ...
41
votes
4answers
956 views

Proving $\sum\limits_{l=1}^n \sum\limits _{k=1}^{n-1}\tan \frac {lk\pi }{2n+1}\tan \frac {l(k+1)\pi }{2n+1}=0$

Prove that $$\sum _{l=1}^{n}\sum _{k=1}^{n-1}\tan \frac {lk\pi } {2n+1}\tan \frac {l( k+1) \pi } {2n+1}=0$$ It is very easy to prove this identity for each fixed $n$ . For example let $n = 6$; ...
40
votes
5answers
2k views

Convergence/Divergence of infinite series $\sum_{n=1}^{\infty} \frac{(\sin(n)+2)^n}{n3^n}$

$$ \sum_{n=1}^{\infty} \frac{(\sin(n)+2)^n}{n3^n}$$ Does it converge or diverge? Can we have a rigorous proof that is not probabilistic? For reference, this question is supposedly a mix of real ...
27
votes
3answers
709 views

$\sum_{n=1}^{\infty}\frac{H_n}{n^2 2^n}=\zeta(3)-\frac{1}{2}\log(2)\zeta(2)$

How can I prove that $$\sum_{n=1}^{\infty}\frac{H_n}{n^2 2^n}=\zeta(3)-\frac{1}{2}\log(2)\zeta(2).$$ Can anyone help me please?
29
votes
10answers
4k views

Direct Proof that $1 + 3 + 5 + \cdots+ (2n - 1) = n\cdot n$

Prove that for all integers $n$, $n \geq 1$, $$1 + 3 + 5 + \cdots + (2n - 1) = n\cdot n$$ How would I go about proving this?
8
votes
2answers
2k views

Nested Radical of Ramanujan

I think I have sort of a proof of the following nested radical expression due to Ramanujan for $x\ge 0$. $$\large x+1=\sqrt{1+x\sqrt{1+(x+1)\sqrt{1+(x+2)\sqrt{1+\cdots}}}}$$ for $ x\ge -1$ I ...
30
votes
2answers
11k views

$\sum k! = 1! +2! +3! + \cdots + n!$ ,is there a generic formula for this?

I came across a question where I needed to find the sum of the factorials of the first $n$ numbers. So I was wondering if there is any generic formula for this? Like there is a generic formula for ...
15
votes
4answers
980 views

Where do summation formulas come from?

It's a classic problem in an introductory proof course to prove that $\sum_{ i \mathop =1}^ni = \frac{n(n+1)}{2}$ by induction. The problem with induction is that you can't prove what the sum is ...
10
votes
2answers
5k views

How do I prove this sum is not an integer

Assume that $k,n\in\mathbb{Z}^+$. Prove that the sum \begin{equation*} \dfrac{1}{k+1}+\dfrac{1}{k+2}+\dfrac{1}{k+3}+\ldots +\dfrac{1}{k+n-1}+\dfrac{1}{k+n} \end{equation*} is not an integer. The ...
13
votes
13answers
4k views

How to explain the formula for the sum of a geometric series without calculus?

How to explain to a middle-school student the notion of a geometric series without any calculus (i.e. limits)? For example I want to convince my student that $$1 + \frac{1}{4} + \frac{1}{4^2} + ...
10
votes
4answers
299 views

Infinite Series $\sum_{n=1}^{\infty}\frac{1}{\prod_{k=1}^{m}(n+k)}$

How to prove the following equality? $$\sum_{n=1}^{\infty}\frac{1}{\prod_{k=1}^{m}(n+k)}=\frac{1}{(m-1)m!}.$$
9
votes
2answers
891 views

If the series $\sum_0^\infty a_n$ converges, then so does $\sum_1^\infty \frac{\sqrt{a_n}}{n} $

Problem: Suppose that for every $n\in\mathbb{N}$, $a_n\in\mathbb{R}$ and $a_n\ge 0$. Given that $$\sum_0^\infty a_n$$ converges, show that $$\sum_1^\infty \frac{\sqrt{a_n}}{n} $$ ...
4
votes
4answers
6k views

How does the sum of the series “$1 + 2 + 3 + 4 + 5 + 6\ldots$” to infinity = “$-1/12$”? [duplicate]

(I was requested to edit the question to explain why it is different that a proposed duplicate question. This seems counterproductive to do here, inside the question it self, but that is what I have ...
1
vote
5answers
488 views

Generate solutions of Quadratic Diophantine Equation

Recently I've asked a question for how to solve Quadratic Diophantine Equation and I got one interesting answer. Link to question: How to solve Quadratic Diophantine Equation Here's the answer: $$ ...
14
votes
2answers
2k views

Do these series converge to the von Mangoldt function?

Jeffrey Shallit formulated this recurrence for me: $\displaystyle T(n,1)=1, k>1: T(n,k) = \sum\limits_{i=1}^{k-1} T(n-i,k-1)-\sum\limits_{i=1}^{k-1} T(n-i,k)$ which is the lower triangular array ...
8
votes
2answers
333 views

recurrence relation $f(n)=5f(n/2)-6f(n/4) + n$

I've been trying to solve this recurrence relation for a week, but I haven't come up with a solution. $f(n)=5f(n/2)-6f(n/4) + n$ Solve this recurrence relation for $f(1)=2$ and $f(2)=1$ At first ...
5
votes
5answers
1k views

The convergence of $\sqrt {2+\sqrt {2+\sqrt {2+\ldots}}}$ [duplicate]

I would like to know if this sequence converges $\sqrt {2+\sqrt {2+\sqrt {2+\ldots}}}$. I know this sequence is increasing monotone, but I couldn't prove it's bounded. Thanks
5
votes
2answers
424 views

If $ x_n \to x $ then $ z_n = \frac{x_1 + \dots +x_n}{n} \to x $

I have this problem I'm working on. Hints are much appreciated (I don't want complete proof): In a normed vector space, if $ x_n \longrightarrow x $ then $ z_n = \frac{x_1 + \dots +x_n}{n} ...
4
votes
6answers
257 views

Find the sum of the series $\sum \frac{1}{n(n+1)(n+2)}$

I got this question in my maths paper Test the condition for convergence of $$\sum_{n=1}^\infty \frac{1}{n(n+1)(n+2)}$$ and find the sum if it exists. I managed to show that the series ...
3
votes
3answers
396 views

Cauchy Sequence. What is this question actually telling me?

Let $(a_n)$ be a sequence such that $\lim\limits_{N\to\infty} \sum_{n=1}^n |a_n-a_{n+1}|<\infty$. Show that $(a_n)$ is Cauchy. So basically I am told that the sum of the difference isn't ...
5
votes
2answers
629 views

Another Epsilon-N Limit Proof Question

How to prove the limit of the following sequence using epsilon-N argument. $$a_n=\frac{3n^2+2n+1}{2n^2+n}$$ I took the limit to be $\frac{3}{2}$ and proceeded with the argument, ...
2
votes
5answers
541 views

Finding generating function for the recurrence $a_0 = 1$, $a_n = {n \choose 2} + 3a_{n - 1}$

I am trying to find generating function for the recurrence: $a_0 = 1$, $a_n = {n \choose 2} + 3a_{n - 1}$ for every $n \ge 1$. It looks like this: $a_0 = 1$ $a_1 = {1 \choose 2} + 3$ $a_2 = {2 ...
1
vote
1answer
517 views

Taylor series for $\log(1+x)$ and its convergence

I know the taylor series of $\log(1+x)$. However I don't understand how to find the convergence for $x>1$ and divergence if $x<1$.
62
votes
4answers
8k views

Motivation for Ramanujan's mysterious $\pi$ formula

The following formula for $\pi$ was discovered by Ramanujan: $$\frac1{\pi} = \frac{2\sqrt{2}}{9801} \sum_{k=0}^\infty \frac{(4k)!(1103+26390k)}{(k!)^4 396^{4k}}\!$$ Does anyone know how it works, or ...
18
votes
2answers
3k views

Limit of Nested Radical $\sqrt{1 + 2 \sqrt{1+3 \sqrt{1+ \cdots }}}$

How does one evaluate show that this limit: $$\lim_{n \to \infty}\sqrt{1 + 2 \sqrt{1+3 \sqrt{1+ \cdots \sqrt{1+(n-1) \sqrt{1+n}}}}}=3$$
12
votes
4answers
2k views

How to solve this sequence $165,195,255,285,345,x$

This is a question appeared in a competitive exam. The question is: Find the unknown term in $165,195,255,285,345,x$ 1)375 $\ \ \ \ \ \ \ \ $ 2)420 3)435 $\ \ \ \ \ \ \ ...
30
votes
3answers
2k views

Why is this series of square root of twos equal $\pi$?

Wikipedia claims this but only cites an offline proof: $$\lim_{n\to\infty} 2^n \sqrt{2-\sqrt{2+... \sqrt 2}} = \pi$$ for $n$ square roots and one minus sign. The formula is not the "usual" one, like ...
23
votes
4answers
836 views

How close can $\sum_{k=1}^n \sqrt{k}$ be to an integer?

How close can $S(n) = \sum_{k=1}^n \sqrt{k}$ be to an integer? Is there some $f(n)$ such that, if $I(x)$ is the closest integer to $x$, then $|S(n)-I(S(n))|\ge f(n)$ (such as $1/n^2$, $e^{-n}$, ...). ...
19
votes
4answers
2k views

What's the value of $\sum\limits_{k=1}^{\infty}\frac{k^2}{k!}$?

For some series, it is easy to say whether it is convergent or not by the "convergence test", e.g., ratio test. However, it is nontrivial to calculate the value of the sum when the series converges. ...
17
votes
3answers
1k views

How to prove Chebyshev's result: $\sum_{p\leq n} \frac{\log p}{p} \sim\log n $ as $n\to\infty$?

I saw reference to this result of Chebyshev's: $$\sum_{p\leq n} \frac{\log p}{p} \sim \log n \text{ as }n \to \infty,$$ and its relation to the Prime Number Theorem. I'm looking into an ...
17
votes
2answers
2k views

Series of logarithms $\sum\limits_{k=1}^\infty \ln(k)$ (Ramanujan summation?)

I had this question earlier, so to say as a "standalone" problem, but now it pops up in context of an analysis with the lngamma-function. As well as we can convert the question of sums of like powers ...
12
votes
3answers
21k views

Problem when integrating $e^x / x$.

I made up some integrals to do for fun, and I had a real problem with this one. I've since found out that there's no solution in terms of elementary functions, but when I attempt to integrate it, I ...
11
votes
2answers
458 views

How to prove that $ \sum_{n \in \mathbb{N} } | \frac{\sin( n)}{n} | $ diverges?

It is stated as a problem in Spivak's Calculus and I can't wrap my head around it.
13
votes
2answers
724 views

Convergence of $\sum_{n=1}^\infty \frac{\sin^2(n)}{n}$

Does the series $$ \sum_{n=1}^\infty \frac{\sin^2(n)}{n} $$ converge? I've tried to apply some tests, and I don't know how to bound the general term, so I must have missed something. Thanks in ...
11
votes
5answers
6k views

Is there any formula for the series $1 + \frac12 + \frac13 + \cdots + \frac 1 n = ?$

Is there any formula for this series? $$1 + \frac12 + \frac13 + \cdots + \frac 1 n .$$
8
votes
5answers
502 views

Limit of $s_n = \int\limits_0^1 \frac{nx^{n-1}}{1+x} dx$ as $n \to \infty$

Let $s_n$ be a sequence defined as given below for $n \geq 1$. Then find out $\lim\limits_{n \to \infty} s_n$. \begin{align} s_n = \int\limits_0^1 \frac{nx^{n-1}}{1+x} dx \end{align} I have ...
7
votes
2answers
6k views

If a series converges, then the sequence of terms converges to $0$.

Following the guidelines suggested in this meta discussion, I am going to post a proposed proof as an answer to the theorem below. I believe the proof works, but would appreciate any needed ...
8
votes
4answers
763 views

Why does this expected value simplify as shown?

I was reading about the german tank problem and they say that in a sample of size $k$, from a population of integers from $1,\ldots,N$ the probability that the sample maximum equals $m$ is: ...
4
votes
6answers
609 views

Sequence sum question: $\sum_{n=0}^{\infty}nk^n$ [duplicate]

I am very confused about how to compute $$\sum_{n=0}^{\infty}nk^n.$$ Can anybody help me?
8
votes
2answers
3k views

Does $\sum\limits_{k=1}^n 1 / k ^ 2$ converge when $n\rightarrow\infty$?

I can prove this sum has a constant upper bound like this: $$\sum_{k=1}^n \frac1{k ^ 2} \lt 1 + \sum_{k=2}^n \frac 1 {k (k - 1)} = 2 - \frac 1 n \lt 2$$ And computer calculation shows that sum ...
7
votes
2answers
235 views

Checking my understanding: $1 - 1 + 1 - 1 + 1 - … = \frac{1}{2}$

I've recently run into a proof that claims that $1 - 1 + 1 - 1 + 1 - 1 ... = \frac{1}{2}$ that proceeds as follows: Let $S = 1 - 1 + 1 - 1 + 1 - 1 + ...$. Then $$S = 1 - (1 - 1 + 1 - 1 + 1 - 1 ...
6
votes
4answers
1k views

Showing inequality for harmonic series.

I want to show that $$\log N<\sum_{n=1}^{N}\frac{1}{n}<1+\log N.$$ But I don't know how to show this.
6
votes
3answers
174 views

Compute the limit $\lim_{n \to \infty} \frac{n!}{n^n}$

I am trying to calculate the following limit without Stirling's relation. \begin{equation} \lim_{n \to \infty} \dfrac{n!}{n^n} \end{equation} I tried every trick I know but nothing works. Thank you ...
-2
votes
1answer
321 views

Find the sum of this series [closed]

Find the value of this series: $$\sum_{i=1}^n \frac{n}{\text{gcd}(i,n)}.$$
63
votes
4answers
3k views

How to sum this series for $\pi/2$ directly?

The sum of the series $$ \frac{\pi}{2}=\sum_{k=0}^\infty\frac{k!}{(2k+1)!!}\tag{1} $$ can be derived by accelerating the Gregory Series $$ \frac{\pi}{4}=\sum_{k=0}^\infty\frac{(-1)^k}{2k+1}\tag{2} ...
41
votes
3answers
1k views

Does the series $ \sum\limits_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}} $ converge or diverge?

Does the following series converge or diverge? I would like to see a demonstration. $$ \sum_{n=1}^{\infty} \frac{1}{n^{1 + |\sin(n)|}}. $$ I can see that: $$ \sum_{n=1}^{\infty} \frac{1}{n^{1 + ...
36
votes
6answers
1k views

A series involving the harmonic numbers : $\sum_{n=1}^{\infty}\frac{H_n}{n^3}$

Let $H_{n}$ be the nth harmonic number defined by $ H_{n} = \sum_{n=1}^{n} \frac{1}{k}$. I'm interested in knowing how to show that $$\sum_{n=1}^{\infty}\frac{H_n}{n^3}=\frac{\pi^4}{72}.$$ I tried ...